Definitely stop off, easy to do. I would guess the gf will appreciate it.
As above a place with secure parking is what you are after ideally with metro nearby – you can buy a book (carnet) of 10 tickets to use as you wish. Paris quite empty at the moment as its holiday season and many locals are away, so roads are quiter than usual. We are in Levallois just outside Periph on West side, last 3 metro stops – don’t know of anywhere here as cheap as McMoonters place with parking though. Remember your priorite a droit here (eg people joing the periph often have right of way and roundabouts in the centre are priorite a droit plus some have traffic lights half way round) and take care generally but locals see a Brit plate and will be more curtious than they would to a French regional plate 😉
Driving into the Periph itsepf is pretty straightforward, once you get inside and off dual carriageways ok just take care as above its quiet at the moment. We allow 3 hours to Eurotunnel and make it easily, we take A16 as we are west side. Note you can put Coquelles in as destination.
I assume you have a decent sat nav (can recommend free waze app) also restaurant app la forchette (the fork) is worth downloading, we got 40% our food last night. Summer street cafe culture is in full swing so enjoy. Ideas: Do the normal arc de tromphe, a bit of the Champs Elyse if only to see the state of the cobbles the TdF race over. Walk from Place de la Corconrde through gardens past Louvre (queue goes quickly) and along river to Notre Dame, gound round the back and onto Il de la Cite (ice cream just over little bridge). Take a river cruise (drinks only, dinner ones rather pricey for so so quality). Pont Alexandre III and Grand Palais, should be some river side bars open. See Eiffel Tower from Trocadero then walk down if you wish. Sacre Couer worth a trip, nice bistro called Cafe Francencouer about 10 mins walk away – lots of classic French movie stairs around that area. Normal tourist advice, watch for pick pockets and “did you drop this” scams.